The physical behavior of gases in highly non-equilibrium states is governed by kinetic equations based on intermolecular interactions. Transport phenomena caused by the excitation of electrons, interactions of gas atoms with photons or fast chemical gas-phase reactions can only be treated with sufficient accuracy by taking the internal energy states of the particles into account. The aim of this project is to investigate the generalization of the Boltzmann equation to systems of particles endowed with internal energy levels. Also the effect of external forces on the colliding particles will be dealt with. We will develop an optimal discretization strategy with respect to the velocity variable for an approximate rigorous solution of the kinetic equations. Thereby, we are faced with the challenge of devising a discretization scheme dealing properly with the presence of electric and magnetic fields. The solution of the extended Boltzmann equation applied to relevant physical problems allows us to analyze relaxation phenomena and the stability of equilibria as well as to study possible transitions to a chaotic dynamics. We intend to derive moment and macroscopic equations of generalized conservation type, and to address the question concerning their closure for an adequate and consistent description. By using kinetic tools, we want to recover rigorously the major physical laws underlying the considered phenomena, as e.g. the mass action law or Planck's law of radiation. not assigned KP: Univ.-Prov. Dr. Gian Luca Caraffini